to represent them by circular arcs. This can always be done where an approximation to the true curve will suffice, and when only a small portion of each curve is used, as is in general the case with wheel-teeth. As substitutes for such portions of the curve suitably chosen arcs of circles having the same curvature are employed: for finding these there are several methods in use.
For set-wheels with cycloidal teeth I have recommended[1] the following method:—A in Fig. 116 is a circular centroid—the pitch circle of the wheel for which teeth are to be constructed,—B its centre,—C and D the centres of two equal describing circles (auxiliary centroids) by which the cycloidal arcs a and b, which it is desired to represent by circles, are drawn; their radius is equal to 0.875 of the pitch. O is the point of contact of the circles A, C and D. Make the angle O C a and O D b = 30°, find the peripheral points a′ and b′ on the auxiliary centroids opposite a and b; draw through a and b a straight line, which it is evident must pass through the point of contact and must therefore be a normal to the elements of the curves at a and b, and join a′ and b′ with the centre B,—then the lines B a′ and B b′ (produced if necessary) cut the normal in the required centres of curvature P and Q. The circular arcs for the faces and flanks of the teeth are then drawn beyond and within the pitch circle respectively, joined together on A to make a fair profile (as in the figure), and repeated symmetrically round the pitch circle.
In the well-known method given by Willis, he attempts to determine the circles best suited for the teeth profiles directly, that is without the use of auxiliary centroids or roulettes. The nature of his approximation, in which he follows out some suggestions
- ↑ Der Constructeur, 3rd Ed. pp. 419.
